&March20, 1956
IONIZATION
1089
CONSTANTS OF N-ACYLAMINO ACIDS
Amorphous to X-ray of the anhydrous and hydrated chromium orthoDry, 200-850" (fine grain) phosphates under both stable and metastable equi- CrP04.6Hz0 --______) 2CrPOd.HzO librium. The suggestion of Ness, Smith and Evans3 that on heating the hexahydrate or the anhydrous Hydothermally Hydrothermally Dry, looo 2000 p.s.i. phase above 200' an oxidation of the chromium oc300-950' 300-950" curs will need re-apprasial. If the formula be written as C r P 0 4 . 0 Ha suggestion of a higher oxidation state appears. On the other hand, it is unlikely Dry, 990" that dehydration to a "semi-hydrate" will cause 2CrP04,HzO P-CrPOd oxidation, especially in view of the fact that a t Hydrothermally, 300-900" higher temperatures the material reverts to the tri- Dry Hydrothermvalent chromium in the orthophosphate, and a t > ally, 300-900" still higher temperatures forms Cr203. The eviDry, >-1175" oc-CrPOI P _I dence shows nothing which contradicts the formulation of this phase as the hydroxy phosphate of Fig. 1. Cr(II1). The powder X-ray data showed that this phase silicate chemistry can be developed since the basic had no relation to any silica forms, nor were any property on which the latter depends is the substiother metastable phases encountered which could tution of A13+for Si4+in tetrahedral coordination. Acknowledgment.-We wish to acknowledge the be regarded as silica mineral analogs. It appears most unlikely that Cr3+ can accept a four-coordi- assistance of AXr.J. J. Comer for his part in obnated position with respect to oxygen and, hence, no taining the electron diffraction patterns. This chromo-silicate chemistry analogous to alumino- work was supported by the U S . Army Signal Corps Engineering Laboratories, Project SC-5594.
lc
I'41
441
~
(3) A. T. Ness, R . E. Smith and R. L. Evans, THISJ O U R N A L , 74, 4685 (1952).
[CONTRIBUTION FROM THE
UNIVERSITY PARK, PA.
DEPARTMENT OF CHEMISTRY, BARNARD COLLEGE,
COLUMBIA UNIVERSITY]
The Thermodynamics of Ionization of Amino Acids. 11. The Ionization Constants of Some N-Acyl Amino Acids1 BY
EDWARD J. KINGAND GRACEW. KING RECEIVED AUGUST23, 1955
The thermodynamic ionization constants of some K-acyl amino acids were obtained from measurements of the electromotive forces of cells with hydrogen and silver-silver chloride electrodes which contained buffer mixtures of the weak acids, their sodium salts and sodium chloride. Measurements were made a t 5' intervals between 5 and 50", and the ionization B CT. The values of the parameters are as follows: A : constants were fitted to the equation -log K = ( A / T ) 1248.54, 1101.03, 908.48, 906.43 and 1279.32; B: -4.8146, -3.7708, -2.8416, -2.9315 and -3.9494; C: 0.014411, 0.012730, 0.011771, 0.012096 and 0.013763 for N-acetylglycine, N-propionylglycine, N-acetyl-m-alanine, N-acetyl-m-aamino-n-butyric acid and N-acetyl-&alanine, respectively. The changes in free energy, entropy, enthalpy and heat capacity associated with the ionization reaction in the standard state as well as the properties a t the temperature a t which the ionization constant reaches its maximum value are compared with those of other carboxylic acids. The entropies of ionization of the acyl amino acids lie between those of the fatty acids and the a-amino acids. This is related in large part t o the orientation of water about the peptide linkage. The entropy of ionization of acetic acid is more negative than that of formic acid because the order-producing methyl group in the former projects into the region of disordered water outside the primary hydration layer about the carboxylate ion. Smaller fluctuations in the entropy of ionization associated with chain branching and lengthening are attributed to the strengthening of the water structure by alkyl groups and to restricted internal rotation in the anions.
+ +
The N-acyl amino acids occupy a strategic position among the carboxylic acids. They are composed of neutral molecules (HA), as are the fatty acids, so that their ionization creates charged particles: HA HzO H30+ A-. Their molecules contain the peptide linkage, -CONH-, found also in peptides and proteins. From that standpoint they are important model compounds to consider before any concerted attack is made on the ionization of peptides themselves. From the theoretical point of view the presence of the peptide linkage is interesting because it is polar, i t will interact with water,2 and it will act as a chain-stiff-
+
*
+
(1) This investigation was supported by a research grant, H-1651, from the National Heart Institute of the National Institutes of Health, U.S. Public Health Service. (2) E. F. Mellon, A . H. Korn and S. R. Hoover, THIS JOURNAL, 70, 3040 (1948).
ener3j4inasmuch as the internal rotation about the carbon-nitrogen bond is severely r e ~ t r i c t e d . The ~ entropy of ionization may be expected on this basis to show some interesting effects. Work is currently in progress in this Laboratory on the ionization of peptides and also of some N-carbamoyl amino acids in which the terminal methyl group of the acetyl derivatives is replaced by the hydrophilic, polar amino group. The ionization constants of the N-acetyl derivatives of glycine, DL-alanine, &alanine and DL-CYamino-n-butyric acid and N-propionylglycine have (3) D. H. Everett, D. A. Landsman and B. R. W. Pinsent, Proc. ROY SOC.(London), 2168, 403 (1952). (4) A. G. Evans and S. D. Hamann. Trans. Faraday Soc., 17, 34 (1951) ( 5 ) S. Mizushima, T. Simanouti, S. Nagakura, M. Kuratani, h l . 7'2, 3490 (1950). Tsuboi, H . Baba and 0. Fujioka, THISJOURNAL,
EDWARD J. KINGAND GRACEw.KING
1090
been obtained from measurements on cells of the types (Pt)H2 IHA(ml), NaA(mz), NaCl(m3)/AgCl-Ag
(I)
Measurements were made a t 5" intervals from 5 to 50' to permit calculation of heats and entropies of ionization and related quantities.
Experimental Apparatus.-The hydrogen supply line was reconstructed largely of one-eighth inch copper tubing with Hoke toggle and metering valves, helium leak-tested, and Swagelok tube fittings. The laboratory barometer was cleaned and recalibrated by the manufacturer after the work on the first acid, acetylglycine. Other features of the apparatus have been described Materials.-Dilute hydrochloric acid and carbonate-free sodium hydroxide solution were made and standardized in the same way as in previous w0rk.O Water used in making these solutions and all others was ordinary distilled water passed through a column of Amberlite MB-1 monobed ionexchange resin. It had a specific conductance of about 0.3 gemmhos. The sodium chloride was the bromide-free sample described before.7 Silvef oxide for the silver-silver chloride electrodes was prepared by slow addition of carbonate-free sodium hydroxide solution to dilute silver nitrate solution. The precipitated oxide was washed repeatedly with water by decantation for a period of eleven days. The dry product was found by flame photometry to contain less than 0.00006% sodium. The N-acetylglycine was a C.P.product (H. M . Chemical Co.) treated with washed Norite and recrystallized three times from water. The N-acetyl-DL-alanine was obtained from Mann Research Laboratories and used without further purification. The acetyl-a-amino-n-butyric acid was also from this source, but i t was treated with Norite, recrystallized twice and air-dried. The N-propionylglycine was synthesized in our laboratory from propionic anhydride and glycine and was recrystallized from water and air-dried. The N-acetyl-P-alanine was supplied by Mann Research Laboratories as a thick sirup. After standing a t room temperature for two weeks this crystallized to a yellowish cake smelling strongly of acetic acid. The compound was very soluble in polar solvents and its solutions supersaturated badly. I t was crystallized from acetone to give white, odorless crystals, m.p. 78.3-80.3'. All of the acids were assayed by titration with standard sodium hydroxide solution. Water was determined by the Karl Fischer method. Various semi-quantitative testsI0 also were made. These results are summarized in Table I . It will be seen that all of the acids are of satisfactory purity except the &alanine derivative. Repeated recrystallization of this pioduct gave no improvement in its assay. The product was examined by paper chromatography for free P-
TABLE I N-Acetyl-
__
DL-LI~.
N-AcetylTest
N-Acetylglycine
DL-
alanine
amino-nbutyric acid
hTN-Acetyl- Propionyl@-alanine glycine
99.8% 99.7% 100.0% 98.4% 99.5% Assay 0.26 0.11 0.02 0.04 0.38 Water ..... ..... ..... ..... .05 Ash < ,004 < ,004 < ,004 < ,004 < ,004 Fe < ,004 < .05 < ,004 < ,004 < .01 c1 < ,004 < ,004 < ,004 < ,004 NHs < .004 < ,004 < ,004 < ,004 < ,004 Po4 < .004 < ,004 < ,004 < ,004 < ,004 H.M. 5 A satisfactory test for free NH3 was not obtained because the compound gave a precipitate with IVessler reagent. (6) H . S. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 2nd ed., Reinhold Publ. Corp., h-ew York, N. Y., 1950, pp. 497-516. (7) E. J. King, THISJOURNAL, 7S, 155 (1951). (8) E. J. King, ibid., 74, 1212 (1952). (9) E. J. King, ibid., 76, 1006 (1954). (IO) M . P. Stoddard and hl. S. D u m , J . Bid. Chem., 142, 329 (1942).
Vol. 78
alanine and none was found although, under the conditions of the experiment, 0.1% could have been detected. Formol titration gave the same assay as that reported below. The product does not contain free amino acid. No evidence that the product was a mixture of acids could be obtained by paper or column chromatography. Anal. Calcd. for CsHgN03: C, 45.80; H , 6.92; N, 10.68. Found: C, 46.0; H,6.5; N, 10.6.11 The performance of our apparatus and technique was checked by making measurements on the cells
Hn IHCl(m) IAgC1-Ag (11) One of us had reported beforeI2that such cells in our laboratory gave higher electromotive forces than those of Harned and E h l e r P above 25". Our present measurements confirm this and point to an explanation. The copper t o mercury contacts of the leads to the hydrogen electrodes are outside the cells and above the water level of the constant temperature bath. The contacts a t the silver-silver chloride electrodes are inside the cells. A thermal electromotive force results which makes the voltage of a cell too high above 25", the maximum effect being 0.11 mv. a t 50°, and too low below 25', the maximum effect being -0.09 mv. a t 5". This is not sufficient to account for the whole of the discrepancy between our results and those of Harned and Ehlers. But the difference in standard potentials a t 25 and 50" ( E 0 2 5 IPSO) = 0.01785 v., obtained by us from measurements on cell I1 is in good agreement with the recent determinations of Bates and Bower14 (0.01785) and Harned and Paxtonl6 (0.01790) and not with that of Harned and Ehlers (0.01802). We conclude that the temperature coefficient of our results, after correction for thermal electromotive force, is satisfactory. The Ea value a t 25O, 0.22258 abs. v., is considerably higher than that of Bates and Bower or Harned and Paxton though reasonably close to that of Harned and Ehlers. This high value may reflect differences in depth of introduction of hydrogen into the solution1eor a decrease in the electromotive forces of the standard cells since their calibration." It has seemed best to minimize these systematic errors by using working standard electro, with our apparatus. These motive forces, E ~ w consistent have been derived by increasing all of the results of Bates and Bower by 0.24 mv. so that E w a t 25' is equal to 0.22258 v., the average of our own results, and by including also the thermal electromotive forces. The values of EOw so obtained are: 5", 0.23429; lo", 0.23160; 15O, 0.22876; 20°, 0.22576; 25', 0.22258; 30°, 0.21930; 35', 0.21594; 40", 0.21240; 45', 0.20868; 50°, 0.20484 abs. v. These values agree with our own limited experimental results on cell I1 with a standard deviation of h0.036 mv. They are used in all of the calculations that follow. The general techniques used in the measurements on cell I have been described before.'-* Solutions were prepared from weighed quantities of acid, sodium hydroxide solution, sodium chloride and conductance water. The addition of the acid and sodium chloride t o the sodium hydroxide and the final weighing were made just before filling the cells. The performance of the cells gives no evidence of hydrolysis of the peptide linkage. The standard deviations in mv. between initial and final readings a t 25' taken about 30 hours apart were 0.053 for acetylglycine, 0.043 for acetylDL-alanine, 0.037 for acetyl-a-amino-n-butyric acid, 0.032 for acetyl-P-alanine and 0.050 for propionylglycine. The electromotive forces, corrected to a hydrogen pressure of one atmosphere, can be represented as a function of temperature by the equation Et = E26 a(t 25) b(t - 25)2 (1) The parameters of this equation for electromotive forces in absolute volts and concentrations in moles per kilogram of
-
+
-
+
(11) Analyses by Schwarzkopf Microanalytical Laboratory, Middle Village 79, N. Y. (12) E. J. King, THISJOURNAL, 75, 2204 (1953). (13) H . S. Harned and R. W. Ehlers, i b i d . , 64, 1350 (1932); 66, 652, 2179 (1933). (14) K . G. Bates and V. E. Bower, J . Research N u l l . Bur. Sfandavds, 53, 283 (1954). (15) H. S. Harned and T. R. Paxton, J . Phys. Chem., 67, 531 (1953). (16) G. J. Hills and D. J. G. Ives, J . Chem. Soc., 305 (1951). (17) G. W. Vinal, "Primary Batteries," John Wiley and Sons, Inc., New York, N. Y., 1030, Chapter 6.
IONIZATION CONSTANTS OF N-ACYLAMINOACIDS
March 20, 1956
water are given in Table 11. The standard deviations between the observed electromotive forces and those calculated from equation 1 are f0.078 mv. for acetylglycine, 0.023 for propionylglycine, 0.025 for acetyl-DL-alanine, 0.024 for acetyl-DL-a-amino-n-butyric acid and 0.030 for acetyl-& alanine.
0,01004 .02003 ,02497 .03001 ,03506 .05015
TABLE I1 EQUATION 1
PARAMETERS OF
ma
Wll
0 0 0 25 0 0
mJmz = 1.168 0.01178 ,02405 ,03348 .04636 ,05654 ,07913 0.01003 .02014 .03022 .04058 .07155 ,1034 0.02006 ,04003 .05980 .07986 .1398 .2002 0.01005 .02060 .03124 .04031 .07473 .loll
0,01004 .02060 .02886 .03966 .04841 .06770
0.55532 ,53634 .52753 .51928 .51417 .50563
0 0 0 0 0 0
496 431 402 374 355 326
Propionylglycine, ml/mz = 0.55628 0.01254 .55357 .02020 .53305 .03026 ,52539 .04067 .51095 .07137 ,50145 .lo34
ml/mz = 2.000 0.01004 0.54472 .52619 ,02018 .51557 .03017 .50827 ,03995 ,06990 ,49384 .48461 .lo03 Acetyl-DL-alanine,m,/m, = 0.01007 0.56177 ,54294 ,02062 .53206 .03124 .04031 ,52553 .50972 ,07474 .1217 ,49719
1.000 498 455 418 392 341 309
-35 -35 -22 -29 -20 -15
461 396 360 335 286 255
-35 -35 -32 -31 -29 -45
1.000 588 520 486 460 410 365
-52 -42 -60 -50 -40 -35
ml/mz = 2.000 0.02066 .04001 .06051 .OS147 .1201 .1994
0.01045 .02000 .03025 ,04088 .06030 .09968
0.54344 ,52617 .51532 .50743 .49740 .48452
Acetyl-DL-a-amino-n-butyric acid, 0.01048 0.01086 0.55996 ,02076 .02152 .54832 .03049 .03924 .52652 ,03584 .03711 .52795 ,04956 .04969 ,52054 0.02076 ,04053 .06019 .07996
ml/mz = 2.000 0.01039 0.54378 .52613 .02027 ,51580 .03009 ,03998 ,50846
Acetyl-@-alanine,mJmz = 1.000 0.01004 0.60423 599 538 ,02017 ,58647 ,02498 .58093 522 .03001 .57635 501 ,03508 .57243 487 .05018 .56351 455
a X 10’ b X 108
Era
Acetylglycine, ml/mz = 1.000 525 0.00836 0.56427 0.00825 ,53975 442 ,02019 ,02107 ,03193 ,03890 .52390 389 ,51708 365 ,04949 ,05015 338 ,07003 .50854 ,07001 .lo087 .lo911 ,49708 294
523 464 429 402 368 325 m1/m2
-56 -48 -50 -42 -45 -45 = 1,000
600 538 485 491 467
-65 -36 -25 -32 -38
541 485 449 426
-45 -62 -45 -42
1091 0 0 20 0 0 0
+
ml/mn =
0.02013 .04027 ,05054 ,06023 .07044 .OS032
0.01006 .02015 ,02529 ,03049 .03544 .04022
2.000 0,58646 ,56869 .56295 .55827 .55448 .55139
542 480 458 443 429 419
- 10
- 15
- 10 - 15 - 15 - 15
Calculations and Results The calculation of the thermodynamic ionization constants of these acids from the electromotive force data was done by the well-known method described by Harned and OwensJ’and will not be described here. The results a t the two different buffer ratios were extrapolated independently to zero ionic strength. The average values of the negative logarithms of the ionization constants obtained in this way are given in Table 111. At the bottom of each column in this table is given the standard deviation of the set of pK values, a measure of the precision of the extrapolations. TABLE I11 THENEGATIVE LOGARITHMS OF THE IONIZATION CONSTANTS O F SOME N-ACYLAMINOACIDS Temp., OC.
5 10 15 20 25 30 35 40 45 50 Stand. dev.
Acetylglycine
3.6816 3.6759 3.6726 3.6671 3.6698 3.6731 3.6779 3.6845 3.6945 3.7063
Propionyl- Acetyl-DLglycine alanine
Acetyl-DLa-amino-nbutyric acid
3.7284 3.7225 3.7184 3.7164 3.7176 3.7208 3.7253 3.7310 3.7402 3.7502
3.6924 3.6943 3.6996 3.7059 3.7158 3.7262 3.7375 3.7502 3.7667 3.7822
3.6992 3.6992 3.7026 3.7075 3.7152 3.7248 3.7334 3.7450 3.7589 3.7735
Acetylalkine
4.4788 4.4652 4.4564 4.4488 4.4452 4.4441 4.4434 4.4452 4.4508 4.4572
0,00073 0.00065 0.00067 0.00083 0.00075
The variation of the ionization constants with absolute temperature can be expressed by equations of the forml8 pK = ( A / T ) B CT (2) All of these constants go through a maximum a t an absolute temperature, 8. Values of this quantity and the corresponding minimum values of pKo as well as the changes in free energy (AFO), enthalpy (AHo),entropy (ASo) and heat capacity (AC,,o) associated with the ionization reactions in the standard state can be derived from the parameters of equation 2 by conventional thermodynamic methods. The constants A , B and C and these various thermodynamic properties are given in Table IV. The only one of these acids whose thermodynamic ionization constant has been reported previ-
+ +
(18) H.S.Harned and R . A. Robinson, Trans. Faraday Soc., 36, 973 (1940).
EDWARD 1.K ~ AND G GRACEw.KING
1092
TABLE IV PARAMETERS OF EQUATION 2 A N D THERMODYNAMIC PROPERTIES ASSOCIATED WITH ACIDS A
Acid
Acetylglycine Propionylglycine Acetyl-DL-alanine Acetyl-DL-or-amino-n-butyric acid Acetyl-&alanine
1248.54 1101.03 908.48 906.43 1279.32
-B C 4.8146 0.014411 ,012730 3.7708 2.8416 ,011771 .012096 2,9315 ,013763 3.9494
ously is a~etylglycine.'~Neuberger found a pK value of 3.684 a t 2.5" as compared with 3.670 reported here. His calculations involve a double extrapolation and it is believed that our result is more trustworthy. The reliability of our results may also be judged by consideration of the sources of error. If random errors of 10.05 mv. in the electromotive forces and standard potentials are assumed and reasonable estimates of the random weighing errors involved in making up the solutions are made, the following minimum probable errors are found : *0.0016 in pK, between hO.0010 and 3t0.0040 in @KO,k 2 . 1 in AF0298, 1 1 6 in AH0298, i.0.06 in ASo298, and * 2 in ACpo2g~.9~20 Systematic errors due to thermal electromotive forces and depth of introduction of hydrogen to the cells will cancel because of the use of EO,. The error due to change in calibration of the standard cells will also largely cancel since the electromotive forces of cells I and I1 do not differ greatly. If the standard cells are off by as much as 0.3 mv., which seems unlikely, the maximum error in pK would be 0.0005. The effect of impurities in the acids must also be considered. If the impurity is inert, either non-acidic or with a very low ionization constant, the error in pK is approximately f(1 7 ) / 7 where f is the fraction of undetected impurity and 7 is ml/m2,the buffer ratio. Thus if 0.1% impurity escapes detection, the error in pK is +0.0020 a t 7 = 1 and +0.0015 a t r = 2. We have made measurements a t two different buffer ratios for each acid and in all cases the two extrapolated values of pK agree within the expected limits. We have tested the effect of impurity in another way. Three buffer solutions were prepared from crude acetyl-DL-a-amino-n-butyric acid (98.97c pure) and measured with the rest. Two of these gave results in good agreement with those based on the repurified material after allowance for the difference in purity; the third deviated by such a large, constant amount as to suggest an accidental error in the preparation of the solution. It is believed that the effect of undetected inert impurities is less than 0.002 in the pK values reported here. None of the systematic errors which have been considered so far would be appreciably temperature-dependent ; none should appreciably affect A H o or ACop. The presence of other acids might cause such an error, This appears to be a
+
(19) A Xeuberger Proc R o y Sac ( L o n d o n ) ,8 1 6 8 , 68 (1937) (20) PI' W Please, Btochem J , 56, 1 Q R (1954) I n Please's notation the variance in # K Ois given by
v(p&)
= [(i/mjp2
+ (T*/h2slj(p + (pz/~h4)(e i)2
-
TI4]V(Yo) where
p =
(T/O) and a
term ( S i / m ) has been neglected
VOl. 78
THE
IONIZATION OF N-ACYL AMINO
AF0m,
AHozas, cal. mole-'
ASQiga, cal. deg.-i mole-'
5006.5 5071.8 5068.3 5069.3 6064.1
-149 -140 -631 - 773 $255
-17.29 -17.49 -19.12 - 19.59 -19.48
cal. mole-'
AC~~roa. cal. des.-' mole-1
-39 -35 -32 -33 -38
e,
OK.
294.3 294.0 277.8 273.7 304.9
9%
3.6690 3.7168 3.6986 3.6909 4.4430
serious possibility only in the case of acetyl-@-alanine. But our experiments prove the presence of no free amino acids and indicate that other acids are probably absent also.
Discussion At any specified temperature the therinodynamic properties are connected by the relation 23026RT($K) = AF' = AH' - TAS"
(3)
For a series of compounds like the fatty acids, the a-amino acids or the N-acyl-a-amino acids pK and A F o show only minor variations from acid to acid. Considerable variations within a series do occur in both A H o and ASO, but they are coupled, large negative values of AHobeing associated with large negative values of TASO so that pK and A F o are almost constant. To assist in placing the data reported in this paper in the proper frame of reference, the free energies, entropies and enthalpies of ionization of various carboxylic acids are shown in Figs. 1-3 as functions of E , the chain length. One purpose of these graphs is to show the effects of chain branching and lengthening in a series of similar compounds like the fatty acids or a-amino acids. For this purpose n is taken as the number of atoms, exclusive of hydrogens and the carboxyl group, in the longest chain of the acid molecule. Many of the acid molecules also contain polar substituents. In these cases groups such as hydroxyl, chlorine, N-acyl, N-glycyl and carboxyl have been counted as contributing one unit to n when they come a t the end of a chain. On this basis, for example, the chain length is taken to be two units for propionic acid, glycine, DLalanine, DL-serine, glycylglycine, chloroacetic acid, the hydrogen malonate ion and all of the N-acyl aamino acids2] The values of the thermodynamic properties presented in Figs. 1-3 are those a t 25". This reference temperature has been chosen because it lies in the middle of the experimental temperature range where the properties are known with the greatest precision. Examination of the corresponding data a t 10 and 50" reveals only minor variation from the relations shown a t 25". Xevertheless it is always well to keep in mind when examining small fluctuations in the thermodynamic properties that the choice of 25' is an arbitrary one. To illustrate, the ionization constants and free energies of ionization of acetyl-DL-alanine, propionylglycine and acetyl-DL-a-amino-n-butyric acid are (21) Similar graphs have been given by Everetta>z*except that n was taken as the total number of carbon atoms in the molecule exclusive of the one in the carboxyl group ( 2 2 ) D H Everett, I i z d chtnz belge, 16, 647 (1951)
March 20, 1956
IONIZATION CONSTANTS OF N-ACYLAMINOACIDS
virtually identical a t 25' but the agreement is less close a t other temperatures. An attempt has been made to indicate in Figs. 2 and 3 the relative reliability of the enthalpy and entropy changes. These estimates, often only rough guesses, have been based in many cases on an examination and sometimes a recalculation of the #original data. Most of the data are from measurements on cells without liquid junction,23some are from measurements which require an extrapolation to eliminate the liquid junction potential,24 .and a few directly measured heats of ionization are i n c l ~ d e d . ~ 5Due attention was paid in estimating the reliability of these data to the inherent precision of the extrapolations and to such details as purity of compounds and exclusion of oxygen. Limitations of space preclude a detailed discussion here. It should be noted carefully that the size of the circles in Figs. 2 and 3 represents more closely an estimate of precision rather than one of accuracy. Probable errors of two or three times the indicated precisions are to be expected in some cases. Nevertheless, in a series of acids all studied in the same way, i t may still be possible to make valid comparisons of one acid with another, for systematic errors will tend to cancel. The thermodynamic property with which we shall be least concerned in the following discussion is the change in heat capacity. The probable error in ACoPis a t least A 2 cal. deg.-' mole.-' and variations in this property from acid to acid are not sufficiently pronounced to permit reliable deductions. The Free Energy and Entropy of 1onization.I n principle, the logical starting point in a search for relations between structure and properties should be provided by statistical mechanics. Let AEOo represent the sum total of contributions to the free energy of ionization resulting from the zero point vibrational energies and such potential energy terms as are independent of temperature. The latter may include the binding energy of the proton (23) Glycine: B. B. Owen, THIS JOURNAL, 66, 24 (1934) and ref. 7, DL-alanine: L. F. Nims and P. K . Smith, J . Bid. Chem., 101, 401 (1933) and P. K. Smith, A. C. Taylor and E. R. B. Smith, ibid., 122, 109 (1937) ; DL-a-amino-n-butyric acid, DL-norleucine, DL-norvaline, a-aminoisobutyric acid, DL-valine, DL-leucine and DL-isoleucine: P. K. Smith, A. C. Taylor and E . R. B. Smith, above; @-alanine: M. May and W. A. Felsing, THIS JOURNAL, 73, 406 (1961); 7-aminobutyric acid: ref. 9; e-aminocaproic acid and glycylglycine: E. R. B. Smith and P. K. Smith, J. Biol. Chem., 146, 187 (1942); DL-serine, DL-threonine, and DL-allothreonine: P. K. Smith, A. T . Gorham and E. R. B. Smith, ibid., 144, 737 (1942); formic acid: H. S. Harned 66, 1042 (1934); acetic acid: H . S. and N . D. Embree. THISJOURNAL, Harned and R. W. Ehlers, ibid., 56, 652 (1933); propionic acid: H . S. Harned and R. W. Ehlers, i b i d . , 65, 2379 (1933); n-butyric acid: H. S. Harned and R. 0. Sutherland, ibid., 56, 2039 (1934); glycolic acid: L. F. Nims, ibid., 6 8 , 987 (1936); lactic acid: L. F. Nims and P. K. Smith, J . B i d . Chem., 113, 145 (1936); chloroacetic acid: D. D. Wright, THISJOURNAL, 66, 314 (1934); hydrogen oxalate ion: H. S. Harned and L. D. Fallon, ibid., 61, 3111 (1939) and G. D. Pinching and R. G. Bates, J . Research Natl. Bur. Sfandards, 40, 405 (1948); hydrogen malonate ion: W. J. Hamer, J. 0. Burton and S. P. Acree, ibid., 24, 269 (1940); hydrogen succinate ion: G. D. Pinching and R. G. Bates, ibid., 45, 322 (1950). (24) Propionic acid, n-valeric acid, n-hexoic acid, isobutyric acid, isovaleric acid, isohexoic acid, trimethylacetic acid and diethylacetic acid: ref. 3 ; 8-chloropropionic acid: E. Larsson, 2. physik. Chem., 8166, 53 (1933). (25) Glycine: J. Sturtevant, THIS JOURNAL, 63, 88 (1941); DLalanine: J. Sturtevant, ibid., 64, 762 (1942) ; @-chloropropionicacid: T. L. Cottrell, G. W. Drake, D. L. Levi, K. J . Tully and J. H. Wolfenden, J . Chem. Soc. (London), 1016 (1948).
7800
'
-
I
QP---dR I /
I
I
1093
'
I
6800
3 5800
-d
E > '0 ca
F;" 4800 Q
3800
2800 0
1
2
3 4 5 6 n. Fig. 1.-The free energies of ionization of carboxylic acids a t 25" as a function of chain length. a-Amino acid hydrochlorides, 0:1, glycine; 2, DL-alanine; 3, DL-a-aminon-butyric acid; 4, m-norvaline; 5, DL-norleucine; 6, 01aminoisobutyric acid; 7 , DL-valine; 8, DL-leucine; 9, DLisoleucine. a-Amino acid hydrochlorides, C ) : (1, glycine) ; 10, @-alanine; 11, y-aminobutpric acid; 12, e-aminocaproic acid. Fatty acids, 0 : A, formic acid; B, acetic acid; C, propionic; D, n-butyric; E, n-valeric; F, n-hexoic; G, isobutyric; H, isovaleric; I, isohexoic; J, trimethylacetic; K, diethylacetic acid. Hydroxy-substituted acids, 0 : 13, DL-serine; 14, DL-threonine; 15, DL-allothreonine; L, glycolic acid; M, lactic acid. Chloro-substituted acids, 9: N, chloroacetic acid ; 0, @-chloropropionic acid. N-Acyl amino acids and peptides, 0 : a , acetylglycine; b, acetylp - a l a h e ; c, propionylglycine; d, acetyl-DL-alanine; e, acetyl-DL-a-amino-n-butyric acid; f, glycylglycine. Anion acids, 0 : P, hydrogen oxalate ion; Q, hydrogen malonate ion; R, hydrogen succinate ion. Solid lines connect acids in which the polar groups, if present, are a t fixed locations. Broken lines connect acids in which the position of the polar group varies.
to oxygen in the acidic hydroxyl group, resonance energy, induction energy and the like.26 The grosser differences in acid behavior have often been interpreted in terms of inductive, polar and resonance effects which are related to these potential energy contributions to A E O O . ~The ~ experimental temperature range is much too far removed from absolute zero to permit the determination of AEOo by extrapolation.28 Consequently, it is nec(26) G. Briegleb, 2 A'alurforsch., 4a, 171 (1949). (27) L. Hammett, "Physical Organic Chemistry," McGraw-Hill Book Co., New York, N. Y., 1940, p. 77. (28) D. H. Everett and W. F. K. Wynne-Jones, Trans. Faraday Soc., 35, 1380 (1939); G. Briegleb, Nalururss., 31, 62 (1943).
EDWARD J. KINGAND GRACEW. KING
1094
Vol. 78
-5
1000
- 10 \
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E
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M
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o
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- 1000 -25
-2000 -30 0
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n. Fig. 2.-The
essary to fall back on the conclusion of Evans and PolanyiZ9that variations in AEoofrom one reaction to another are more closely matched by variations in AF'T than by those in AH'T. From this point of view the acyl a-amino acids, all of which have about the same value of the free energy of ionization a t 25', should have the same value of AEoo. This is reasonable since the location of the polar peptide linkage with respect to the carboxyl group is the same in all four of these acids. By contrast, acetyl-/?-alanine is a weaker acid with values of AFo298and AEOo nearer those of the unsubstituted fatty acids because the peptide linkage is further removed from the ionizing group. That the free energy, and not the heat of ionization, is more closely related to AEoocan be illustrated by the observation that the heats of ionization of the fatty acids and comparable acyl amino acids are almost identical. The higher heat of ionization of acetyl-/?-alanine as compared with that of acetylglycine (a and b in Fig. 3) is to be expected because of the removal of the peptide linkage further from the carboxyl group. Comparable increases in heat of ionization are found between chloroacetic and P-chloropropionic acids (N and 0) and between hydrogen malonate and hydrogen succinate ions (Q and R). Up to this point the effect of molecular motions as expressed in partition functions has not been considered. The entropy of ionization is of particular interest in this connection since i t does not depend on AEoo. A successful treatment of the en( 1 9 ) h i . G Evans and M Polanyi, Tvans F a r a d a y Soc , (1'336)
s%, 1333
I
I
0
1
2
3
4
I
I
5
6
It.
Fig. 3.-The
entropies of ionization of carboxylic acids a t 25' as a function of chain length.
I
enthalpies of ionization of carboxylic acids a t 25' as a function of chain length.
tropies of some neutral molecules, monatomic ions, and roughly spherical oxyanions has been based on partition functions for a smoothed-potential model of l i q ~ i d sbut , ~ the ~ ~ applicability ~~ of this model to the complex acids under consideration is open to question. In any case, owing to cancellation of contributions from the un-ionized and ionized forms of the acid in computing the entropy of ionization, the conventional contributions due to translation, rotation, vibration and free volume probably do not produce fluctuations of ASo for a series of similar acids exceeding 0.2 cal. mole-' deg.-I. Almost all of the entropy of ionization must arise from interactions between solute and solvent which lead to special orientations of solvent molecules about solute particles and also to restrictions on the internal rotations of these particles. That an adequate treatment of the free energy of ionization is sometimes possible without a detailed description of solute-solvent interaction is illustrated by the success of the electrostatic theory of Kirkwood and W e ~ t h e i m e r . ~Estimates ~ of the entropy of ionization based on this theory are far too low, and Kirkwood and Westheimer have suggested that this is due to their neglect of the effect of the ions on the structure of water.33 Though the molecular shapes of the acyl amino acids are not favorable ones for a quantitative treatment using their model, the (30) R. E. Powell and TI'. M. Latimer, J . Chem. Phyr., 19, 1139 (1951). (31) R. E. Powell, J . Phys. Chem., 58, 528 (1954). (32) J. G. Kirkwood and F. H. Westheimer, J . Chem. Phys., 6, 506, 513 (1938); F. H. Westheimer and M. H. Shookhoff, THISJ O U R NAL, 61, 555 (1939). Y (33) F. H. Westheimer and J. G. Kirkwood, T m i i s . F Q V Q ~ QSoc., 43, 77 (1947).
March 20, 1956
IONIZATION CONSTANTS OF N-ACYLAMINOACIDS
method of Kirkwood and Westheimer could provide, in principle, a reasonable point of view for an explanation of the lower free energies of ionization of the acyl amino acids as compared with the fatty acids. I n considering entropies of ionization we are, of necessity, reduced to more qualitative considerations. Water Orientation Effects.-The orientation of water about simple, spherical ions gives rise t o a compressed sheath of water with lower ent r ~ p y . ~ Outside ~ - ~ ~ of this primary hydration sheath a disordered or depolymerized region of abnormally high entropy probably occurs until a t larger distances from the ion the normal loosepacked, semicrystalline structure of water is re-establi~hed.~ The ~ solvation of simple ions is accompanied by a rough coupling of heat and entropy effects, ;.e., large negative heats of solvation are associated with large negative entropies of solvation.39 Much less is known about the arrangement of water molecules about a planar, hydrogen-bonding group like the carboxylate ion, but i t certainly produces a strong ordering of neighboring water molec u l e ~ .The ~ ~effective ~ ~ ~ range of influence of the carboxylate and ammonium ion! on the structure of water appears to be about 5 A.3,42 This would place the peptide linkage in the acyl a-amino acids well within the sphere of influence of the carboxylate ion while the peptide linkage of acetyl-P-alanine would be near the outer limit of this influence. Some orientation of water molecules about uncharged carboxyl, hydroxyl and amino groups is to be expected because of hydrogen bonding, but this is apparently much weaker than that about charged groups.41 Many of the more marked differences in entropy of ionization have been explained on the basis of preferential solvent orientation about charges with consequent loss of e n t r ~ p y . ~The . ~ ~ionization of fatty acids, for example, is ionogenic and produces a large negative entropy of ionization because of the strong orientation of water about the ions. In the ionization of a-amino acid hydrochlorides the charged ammonium group binds water both to the un-ionized and to the ionized forms so that less water can be influenced by the carboxylate ion and the entropy of ionization is considerably less negative than that of a fatty acid. The peptide linkage in the larger solid glycyl peptides has been found to bind waterj2but adsorption of water by the smaller crystalline peptides is probably prevented by saturation of the hydrogen-bonding power of the groups in the crystal. There is no reason to expect such saturation to occur in solution. Orientation of water about the peptide linkage should be aided by slight charges, roughly four-tenths of a unit (34) J. D.Bernal and R. H. Fowler, J. Chem. P h y s . , 1 , 515 (1933). (35) D. D. Eley and M. G. Evans, Trans. Faraday Soc., 34, 1093 (1938). (36) D.H.Everett and C. A. Coulson, ibid., 36, 633 (1940). (37) E. J. Verwey, Rec. frav. chim., 61, 127 (1942). (38) D.D.Eley, Trans. Faraday Soc., 40, 184 (1944). (39) H.S.Frank and M. W. Evans, J. Chem. Phys., 13, 507 (1945). (40) R. G. Gurney, “Ionic Processes in Solution,” McGraw-Hill Book Co., New York, N. Y., 1953,pp. 169,176. (41) G. H. Haggis, J. B. Hasted and T. J. Buchanan, J. Chem. Phys.. 20, 1452 (1952). (42) D.H.Everett and B. R. W. Pinsent, Proc. R o y . Soc. (London), !USA, 416 (1952).
109.5
charge,43borne by the oxygen and nitrogen. Water bound to the peptide linkage is within 5 8. of the carboxylate group and it is therefore reasonable to expect the entropy of ionization of acetylglycine or propionylglycine to be less negative than that of propionic acid but a good deal more negative than that of glycine, as is indeed the case. Examination of Fig. 2 reveals that a number of acids have entropies of ionization which are 5 cal. deg.-’ mole-’ less negative than those of the fatty acids: chloroacetic acid (ASOZSS= - 16.97), glycolic acid (-16.94)) acetylglycine (- 17.29), propionylglycine (- 17.49) and formic acid (- 17.24). Chloroacetic and glycolic acids, like the two acyl amino acids, have strongly polar groups attached to the a-carbon atom. The direct electrostatic effect of these on the carboxyl group is largely exerted within the molecule itself and not through the solvent.32 Contributions to the entropy from this source are therefore probably small and solvent orientation about the polar groups must be largely responsible for the small entropies of ionization of these acids as compared with the fatty acids. Some difference in entropies of ionization is to be expected also because of restricted rotation. In the fatty acids the internal rotations of the alkyl chain are restricted after i ~ n i z a t i o n . ~I n the polar acids under consideration considerable restriction of rotation may exist already in the un-ionized form owing to a combination of steric and electrostatic effects, with the result that less entropy is lost on ionization. Such effects, which probably amount to no more than one entropy unit, will be discussed in the next section. A further example of a change in entropy of ionization of about 5 units by introduction of polar groups is noted in comparing succinic acid (ASO298 = - 16.68 cal. deg.-’ with d-tartaric acid ( - 11.40).45 On the other hand, the five unit differences in entropies of ionization of glycine and p-alanine (1 and 10 in Fig. 2) and of hydrogen oxalate and hydrogen malonate ions (P and Q) are clearly reflections of differences in charge interaction and solvent orientation effects arising from increasing the distance between charged groups. Formic acid occupies a special position. I t is instructive to compare the entropy changes in the reactions
+ HCOO- +HCOOH + CHsCOO-; = -4.9 cal. deg.-l mole-’ ( A ) CHsiYHs+ + NHa +XHc+ + CHsNH2; CHsCOOH
ASo298
ASO~SS= -3.7 cal. deg.-l mole-’
(B)
The two entropy changes are of the same order of magnitude and might, indeed, be still closer together if contributions from moments of inertia and restricted internal rotations of the methyl groups in the acetate and methylammonium ions were taken into account.46 It is difficult to see how these (43) G. B. Carpenter and J. Donohue, THISJOURNAL, 72, 2315 (1950). (44) G. Pinching and R. G. Bates, J. Research N a l l . Bur. Slandards, 46, 444 (1950). (45) R. G. Bates and R. G. Canham, ibid., 47, 343 (1951). (46) The entropy change of -3.7 units in reaction B is the observed entropy change (R. G. Bates and G. D. Pinching, i b i d . , 42, 419 (1949); D. H.Everett and W. F. K. Wynne-Jones, Proc. Roy. SOC.(London), 1778, 499 (1941)) increased by 0.57 unit to correct for the different symmetry numbers.
lOYG
EDWARD J. KING AND GRACE A’. KING
entropy changes can have a common cause and particularly a cause originating in the same kind of ionsolvent interactions, for the comparable ions come on opposite sides of the two equations. The entropy change in the second reaction has been attributed to a solvent exclusion effect which makes the entropy of the methylammonium ion less negative than that of the ammonium ion.4t47 Alternatively, this can be expressed as a decrease in hydrogenbonding power of the ammonium ion by introduction of the methyl group.4* But solvent exclusion would tend to make the entropy change of reaction A positive. Entropy effects associated with restriction of rotation of the methyl groups in the acetate and methylammonium ions would also be of the opposite sign. An explanation of the similar entropy changes of reactions A and B must be based upon a more definite picture of the structure of water about alkyl groups and the carboxylate and ammonium ions. That alkyl groups have an effect on the structure of water is shown by the large decrease in entropy and increase in heat capacity associated with the introduction of the smaller saturated hydrocarbons in water.39 The orientation of water about alkyl groups, which gives rise to a restricted librational. motion of the water molecules30and loss of entropy, may resemble the arrangement of water in the solid hydrates of the hydrocarbon^.^^ These structures, which are consistent with the X-ray diffraction data,50are open patterns of water molecules with only slight distortion of the hydrogen bonds from the tetrahedral angle. The roughly spherical ammonium ion is surrounded by a primary hydration shell containing about four tightly held water molecules.41 The replacement of one hydrogen of the ammonium ion by a methyl group displaces a water molecule from this first hydration shell, increases the entropy of the hydrated ion and makes the entropy of ionization of the methylammonium ion more negative than that of the ammonium ion itself. If the methyl group is replaced by an ethyl group, the entropy of ionization is found to be less negative. This can occur if the outer end of the ethyl group penetrates the envelope of disordered water molecules39outside the primary hydration shell of the ion. The alkyl group through its displacement of high-entropy water and its structure-strengthening ability will decrease the entropy of the ion. For the same reason the entropies of ionization of the di- and triethylammonium ions are less negative than those of the corresponding methyl derivatives. The planar, triangular-shaped carboxylate group will certainly not give rise to the same orientation of solvent as the ammonium ion does. The carboxylate group probably binds about six water molecules in the primary hydration shell. 4 1 These water molecules can be arranged so that the “back” of the anion is outside of the tightly bound pri(47) G. Briegleb, 2. Eleklrochem., 5 3 , 350 (1949). (48) A. F. Trotman-Dickenson, J . C h e m . SOC. ( L o x d o x ) , 1296 (1949). (49) W. F. Claussen, J . C k c m . Phys., 19, 662, 1425 (1951); W. F. Claussen and M. F. Polglase, THISJOURNAL, 74, 4817 (1952). ( 5 0 ) M . von Stackelberg aQd H. R.Muller, J . Chem. P k y s . , 19, 1319
(1351?,
VOl. 7s
mary layer. Displacement of solvent from this layer would then not occur as a result of introduction of a methyl group. It is reasonable, on the other hand, to suppose that the outlying disordered region of solvent would extend around the hydrogen in back of the formate ion. The methyl group in the acetate ion would then protrude into the disordered region rather than into the primary hydration shell. This will make the entropy of the acetate ion more negative than that of the formate ion because the methyl group displaces water from the disordered region and reduces its positive contribution to the entropy and the methyl group by virtue of its structure-strengthening ability causes an ordering of water molecules in the disordered region with a decrease in their entropy. Hence, the entropy of ionization of acetic acid will be more negative than that of formic acid. The formate ion from this point of view is less orderproducing than the acetate ion and this is consistent with its higher m0bi1it.y.~‘ With the propionate ion the ethyl group will penetrate further into the disordered region and give rise to a still more negative entropy of ionization. If the rangeaof influence of the carboxylate group is about 5 A.3 and if the degree of disorder is greatest near the primary hydration layer, it is logical to expect the increase in size of the alkyl group from methyl to ethyl to cause a considerably smaller change in entropy of ionization than the introduction of the methyl group in the first place and this is the case. It is thus possible to offer a plausible explanation of the similar entropy changes of reactions A and B , though a different type of ion-solvent interactim must be involved in each case. 4 similar interpretation of the changes in heat capacity associated with these two reactions is also possible, but the experimental uncertainty in these quantities is so large that a detailed discussion is hardly worthwhile. Another feature of this picture is less satisfying. The small change in heat of ionization between formic and acetic acids does not seem consistent with the close coupling of heat and entropy changes for the solvation of hydrocarbons or other nonpolar molecules like the noble gases.39 In this respect the transition from formic acid to acetic acid is different from the transitions from straight chain to branched chain or short chain to long chain acids discussed in the next section. That this necessarily indicates a different origin of the entropy changes for the latter transitions can hardly be proved or disproved until specification of solute-water interactions can be made more precise and quantitative. In the treatment of chain length and chain brancliing effects in the next section we shall consider not only water-alkyl group interactions of the type described above but also the possible effect of restricted rotation of the alkyl g r o u p ~ . ~ ~ ~ ~ * ~ Chain Branching and Lengthening Effects.4 number of fairly small entropy effects are associated with branching and lengthening of alkyl chains in homologous acids. The replacement of a hydrogen atom on the a-carbon atom by a methyl group invariably results in a decrease of from 1 to 2 cal. mole-’ deg.-l in the entropy of ionization. This is (51) Ref 40, p 201.
March 20, 1956
IONIZATION
CONSTANTS OF N-ACYLAMINOACIDS
illustrated by the transitions from propionic acid to isobutyric acid (C to G in Fig. 2 ) , glycolic acid to lactic acid (L to M), glycine to a-alanine (1 to 2 ) and acetylglycine to acetyl-DL-alanine (a to d). Substitution of a second methyl group for hydrogen on the a-carbon atom causes a further but smaller decrease, e.g., isobutyric acid to trimethylacetic acid (G to J) and a-alanine to a-aminoisobutyric acid ( 2 to 6). Chain branching farther away from the carboxyl group is associated with still smaller changes in entropy. Lengthening of the alkyl chain also is accompanied in some cases by entropy decreases. The effect is most noticeable when the chain length is increased from two to three carbon atoms, e.g., propionic acid to butyric acid (C to D), isobutyric acid to isovaleric acid (G to H), alanine to a-amino-n-butyric acid ( 2 to 3), a-aminoisobutyric acid to valine (6 to 7) and serine to threonine or allothreonine (13 to 14 or 15). These effects are closely associated with the presence of charged groups. When chain branching or lengthening occurs beyond 5 8.of the carboxylate group, little or no change in the entropy of ionization is found, e.g., isohexoic vs. n-valeric acid (I and E). The ionization of the ammonium group in amino acids, a reaction which involves the loss of ionic charge, is associated with positive chain-branching effects in contrast to the negative ones described above for reactions in which charge fields are created. For both the chain-branching and chain-lengthening effects there is a coupling of heat and entropy changes. If, for example, branching a t the a-carbon atom causes a more negative value of T A P , this is associated with a more negative value of AHo,so that AFois affected only slightly. The coupling is closest for a given chain length, n, and is particularly close in the a-amino acid series where the presence of the charged ammonium group in both the un-ionized and ionized acid probably swamps minor variations. The coupling is least evident for the increase in chain length from n = 2 to 3; this is almost invariably associated with a small decrease in AFO. These couplings of heat and entropy changes are reminiscent of the linear relation between heats and entropies of solvation of gaseous atoms, molecules and ions.39J2@ The origin of these effects can be sought in solvent orientation and restricted rotation. It has been suggested that the solvent exclusion effect is responsible for the entropy changes due to chain branching,22but this cannot be an important factor since it would tend to make the changes less negative whereas more negative values of As0 are observed. The order-producing interaction of alkyl groups with water discussed in the preceding section would also produce less negative entropies of ionization by chain branching if the alkyl groups were within the innermost hydration shell about the carboxylate ion. If it be supposed, as was done in the discussion of formic and acetic acids, that this shell does not extend in back of the ionic group, then the alkyl groups may lie in the disordered or depolymerized region outside this innermost shell. ( 5 2 ) R. P. Bell, Trans. Faraday Soc , 5 5 , 496 (1937). (53) I. hl. Barclay and J. A. V. Butler, rbid., 34, 1445 (1938).
1097
This is certainly possible if the alkyl group branches from the a-carbon atom. Even if the primary hydration shell does extend back to some extent so that the disordered region starts as far back as 2.8 A., the oxygen-oxygen distance in ice,39a methyl group on the a-carbon atom could still partially project into the disordered region. Now the disordered region is created in the ionization process, and the alkyl group because of its structurestrengthening ability will counteract this disorder making the positive contribution of the disordered region to the entropy of the ion smaller. The entropy of ionization of a branched chain acid should thus be more negative than that of an unbranched acid. Replacement of an a-methyl group by ethyl should cause greater penetration of the disordered region and a further decrease in the entropy of ionization. This change will be smaller than that attending the original introduction of the methyl group because the degree of disorder is probably greatest nearest the primary hydration shell. Thus the entropy of ionization of acetyl-m-a-amino-nbutyric acid is only 0.47 cal. mole-‘ deg.-l more negative than that of acetyl-DL-alanine whereas the latter is 1.8 units more negative than that of acetylglycine. The close coupling of heat and entropy changes associated with chain branching is consist ent with the coupling of these changes for solvation of the simple hydrocarbon^.^^^^^^^^ Our knowledge of the arrangement of water molecules about a carboxylate ion is not sufficient to enable us to push this description further. Another possible cause of the anomalous chain branching and lengthening effects can be sought in restrictions on internal rotations that arise from the creation or disappearance of charged groups by the ionization r e a c t i ~ n . ~ .With ~ , ~ ~gaseous and liquid phases it has generally been found that steric repulsion is the primary cause of restricted rotation and electrostatic forces have only a secondary eff e ~ t .HilP ~ ~ has made some calculations of the influence of ionic charges on internal rotations of solutes in aqueous solution based on electrostatic interaction through a structureless continuum. He concluded that the effect was of secondary importance though it might approach in magnitude the non-electrostatic contribution to restricted rotation in certain cases if local variations in the effective dielectric constant a t different orientations about the ion were considered. But charged groups may cause restriction of rotation in a different way : the compressed, tightly bound water about charged groups, whether in the primary hydration layer or in the outer disordered zone, could restrict rotation of alkyl groups imbedded in this r e g i ~ n . ~In , ~view of our lack of detailed knowledge of the structure of water and the nature of restricted rotation, quantitative calculations can hardly be made. But the hypothesis that restricted rotation is responsible for the chain lengthening and branching effects can be tested by rough estimates and qualitative comparison with the data. The effect of substitution of methyl for hydrogen (54) S. Mizushima, “Structure of Molecules and Internal Rotation,” 1954,Chapters 111,I V and VI. Academic Press, Inc., New York, N.Y., ( 5 5 ) T. Hill, J . Chem. P h y s . , 11, 545, 552 (1943); 12,56,147 (1944); THISJOURNAL, 75, 5304 (1951).
1098
EDWARD J. KINGAND GRACEW. KING
on the a-carbon atom is a test case. The methyl group is well within the 5 A. limit of influence3 of the carboxylate group on the water structure. It is thus subject in the anion of the acid to restriction both from direct electrostatic interaction and from being imbedded in a rigid water sheath about the charged group. Consider, as a crude approximation, the methyl group of acetyl-DL-alanine to be a symmetrical top affixed to a rigid framea56 Then with reasonable guesses for the restricting potentials (1 to 4RT before ionization and 8 to 16RT afterwards) it is easy to predict an entropy decrease accompanying ionization due to restricted rotation of 1 to 2 cal. deg.-l mole-'. This is comparable with the 1.8 unit difference between the entropies of ionization of acetylglycine and acetyl-DL-alanine. The contribution of restricted rotaton to ACo, is small and probably positive, that to AHo is negative. Both effects are of the correct sign but are smaller than the experimental data require. The coupling of heat and entropy changes is not close, so that the contribution of restricted rotation to AFO is larger than that observed, The crudities of the model must prevent us from placing too much weight on these calculations, but they do indicate that restricted rotation can give rise to contributions to the thermodynamic properties which are in the right direction and the effect on ASo is of the right order of magnitude. Other observations are consistent with the hypothesis. Chain branching on carbon atoms further removed from the carboxylate group leads to smaller effects as the methyl group is withdrawn from the tight water sheath about the charge. Substitution of an ethyl group for methyl on the acarbon atom results in a still more negative entropy of ionization, because the ethyl group has more rotational degrees of freedom to be restricted. For this reason the entropy of ionization of acetyl-aamino-n-butyric acid could be and is more negative than that of acetylalanine. The changes in entropy of ionization associated with lengthening of the alkyl chain in substituted ammonium ions and fatty acids have been attributed to restriction of rotation about carbon-carbon bonds in the chain caused by the tight binding of water about the charged g r o ~ p . ~In, ~acids with a chain three carbon atoms long, rotational isome r i ~ mabout ~ ~ the a- and p-carbon atoms should be possible. If ionization stabilizes one of the isomers, perhaps the trans configuration, loss of entropy associated with the mixture of isomers could result. This is a possible explanation for the more negative entropies of ionization of acids with n = 3 as compared with acids with shorter chains. The acyl a-amino acids are interesting in this connection because of the stiffness of the peptide linkage.5j43.54s57The two carbonyl groups are frozen in the trans configuration regardless of the state of ionization of the acid. Since there remain fewer rotational modes to be frozen out by ionization, these acids should have less negative entropies and enthalpies of ionization than do the long chain ( 5 8 ) K. S. Pitzer, "Quantum Chemistry," Prentice-Hall, New York, N. Y.,1963, Appendix 18. (57) W. D. Phillips, J . Chew Phys.,23, 1383 (1055).
Inc.,
Vol. 78
fatty acids. A possible stabilization by ionization of one of the rotational isomers about the aand 0-carbon atoms in acetyl-&alanine could make its entropy of ionization more negative than those of acetyl- or propionylglycine. None of these contributions to the entropy of ionization due to restricted rotation should exceed 2 cal. deg.-' mole-l and contributions of 1 unit or less are most likely. They cannot be separated from water-orientation effects which they will tend to reinforce. The Temperature of Maximum Ionization Constant.-The ionization constants of practically all carboxylic acids reach maximum values a t temperatures between 0' and 60'. The absolute temperature, 0, corresponding to this maximum is related to AHoa t a specified temperature T by A H % = 2.3026RC(e2 - T2) (4) The constant C is a parameter of equation 2 and is related to ACO,. Since C does not vary much from acid to acid, it follows that a t any specified temperature, e.g., T = 298.16'K., variations in AHo will be matched roughly by corresponding ones in The values of 0, like those of AHO298, are about the same for comparable fatty acids and acyl amino acids. Chain branching a t the alpha-carbon atom in either series produces a drop of 15 to 20' in 0. This large change is all the more remarkable when i t is noted that the difference in 8 values of a fatty acid and an alpha-amino acid, two acids of different charge type, is only about 30'. Gurney69has developed an expression for 0 on the assumption that the work required to remove a proton from the acid molecule is separable into a part independent of temperature (Jnon) and an electrostatic contribution (J,l) which varies with the reciprocal of the dielectric constant of water
e
=
k[i
+ ( J ~ ~ ~ / J ~ ~ ( ~ ) ) I( 5 )
where k is a constant characteristic of the pure solvent. However successful this approach may be for a description of the behavior of acids of radically different type, it is manifestly inadequate as the basis for interpreting differences in 0 values of similar acids. It does not seem reasonable, for example, to try to account for the large change in 8 produced by introduction of a methyl group on the a-carbon atom by either an increase in J,I or a decrease in JnOn. All expressions, including equation 5, which attempt to abbreviate the contributions of solutesolvent interaction to a simple electrostatic effect through a structureless dielectric are inadequate for the treatment of similar acids. A more explicit accounting of water orientations, including their effect on restricted rotation, is required. The importance of solvent orientation may be illustrated by the observation that the difference in temperatures of maximum ionization constant between acetic acid in water and deuteroacetic acid in deuterium oxidec0is the same, within the experimental error, as the difference in the temperatures ( 5 8 ) A similar prediction can be made when the ionization constants are fitted t o the equation18 log K = ( A / T ) B log T C: in t h a t case AHOT = R B ( T - 8 ) . (59) Ref. 40, pp. 128-132. (80) F Brescia, V. K . 1,aMer and F. C. Nachod, THISJ O U R N A L , 62, 614 (1940).
+
+
March 20, 1956
FREEDIFFUSIONI N %COMPONENT SYSTEMS WITH
INTERACTING
FLOWS
1099
of maximum density of the pure solvents. The lat- chain branching and lengthening effects all but ter property is an indication of the relative struc- disappear when entropy changes are compared a t .'8 At 298.16'K. the entropy of ionization of tural strengths of the two liquids. The suggestion has been made by Harned and acetyl-DL-alanine is 1.8 units more negative than EmbreeG1that comparisons of acid strength should that of acetylglycine. At 277.4'K. the temperabe made a t 0 rather than a t some arbitrary temper- ture of maximum ionization constant for the ature like 298'. There appears to be no advantage branched chain acid, the ice-like ordered structure in this so far as PKe values are concerned, and com- of the solvent predominates and less change in oriparisons of AFOo = -2.3026Rt?(pKe) are not a t all as entation of water can be produced as a result of favorable as those a t some constant temperature ionization. The entropy of ionization of acetylalabecause of the variable factor 0. The entropy nine a t this temperature therefore is less negative change a t 0 shows the same behavior as PKe (or than that a t 298.16' and differs from that of acetyleven pK298) since ASo@ = 2.3026R(pKs).60 Thus glycine a t its temperature of maximum ionization (294.3') by only 0.13 unit. (61) H. S. Harned and N. D. Embree, THIS JOURNAL, 66, I050 NEWYORE,.'uT Y. (1934).
[CONTRIBUTIONFROM THE DEPARTMENT O F CHEMISTRY, UNIVERSITY
OF
WISCONSIN]
An Exact Solution of the Equations for Free Diffusion in Three-component Systems with Interacting Flows, and its Use in Evaluation of the Diffusion Coefficients BY HIROSHIFUJITA~ AND LOUISJ. GOSTING RECEIVED SEPTEMBER 26, 1955 A rigorous solution of the differential equations for one-dimensional free diffusion is obtained for a three-component system in which the solute flows interact; the only assumptions are that the volume changeon mixing and the concentration dependence of the diffusion coefficients are negligibly small. Based on this solution, a general procedure is developed for computing the four diffusion coefficients from data for the reduced height-area ratios, DA, and reduced second moments, DW,of the refractive index gradient curves of two or more experiments. An additional procedure for calculating the coefficients is devised which depends primarily on measurements of DAand the fringe deviation graphs obtained with the Gouy diffusiometer. These procedures are applied to recently reported data for free diffusion in aqueous solutions of mixtures of (1) LiCl and KCl and (2) LiCl and NaCl. New values for the four diffusion coefficients of each system are reported and compared with those obtained using the previous methods of calculation.
I n general, the diffusion of either solute in a solution containing three components depends on the concentration gradients of both solutes, four diffusion coefficients being required to describe the system. This interaction or coupling of flows is conveniently represented by a modified form2pa of Onsager's4 phenomenological flow equations. Data illustrating such interaction in two electrolyte systems have recently3 been obtained using the Gouy diffusiometer. I n recent papers two procedures were developed for computing the four diffusion coefficients, but they are subject to certain limitations. A general method2 utilizing reduced second and fourth moments, azm and D:m, of the refractive index gradient curves from two experiments is limited because of experimental inaccuracy in the reduced fourth moments. A second procedurea in which values of the diffusion coefficients are selected to reproduce best the observed fringe deviation graphs and reduced height-area ratios, a ~of ,a t least two experiments gives good results when i t is applicable; however, i t is based on series expansions for the concentration curves which are applicable only when one cross-term diffusion coefficient is sufficiently small. (1) On leave from the Department of Fisheries, Faculty of Agriculture, Kyoto University, Maizuru, Japan. (2) R. L. Baldwin, P. J. Dunlop and L. J. Gosting, THIS JOURNAL, 77, 5235 (1955). (3) P. J. Dunlop and L. J. Gosting, ibid., 7 7 , 5238 (1955). (4) L. Onsager, A n n . N. Y. Acad. Sci., 46,241 (1945).
Using the exact solutions derived below for the solute concentration distributions in free diffusion, new procedures are devised for calculating the four diffusion coefficients. The general method utilizing values of ai)^ and anmshould be more accurate than the earlier procedure utilizing a 2 m and Dim, because DA can be measured much more accurately than aim. The new second method, which depends primarily on values of DAand the fringe deviation graphs, has the advantage that neither cross-term diffusion coefficient need be small.
Theory Basic Equations.-The equations for one-dimensional diffusion in a three-component system are written5
in which solute concentrations CIand CZare functions of position x and time t, Dll and D,,are the main diffusion coefficients, and D ~and Z D2l are the cross-term diffusion coefficients. For free diffu(5) These equations are obtained by substituting the Bow equations 1 and 2 of ref. 3 into the continuity equations. It is here assumed that the diffusion coefficients are all independent of concentration and t h a t no volume change occurs on mixing. These conditions may be approached experimentally by making the concentration differences across the initial boundary sufficiently small. The reader is referred to refs. 2 and 3 for a more detailed description of the flow equations and the conditions under which they are valid.
